nLab
Morse-Kelley set theory

The Morse–Kelley set theory (MKMK) is an axiomatic approach to set theory which has both classes and sets. Whereas NBG (which also has both classes and sets) is conservative? over ZFC, Morse–Kelley is not a conservative extension of NBGNBG. The principal difference from NBGNBG is that MKMK allows arbitrary formulas ϕ\phi appearing in the class comprehension axiom schema (in particular, formulae with quantifiers ranging over classes themselves).

The approach is explained in the appendix to John Kelley’s 1955 book General Topology.